1

CalibrationofMultifactorHestonModels

toCreditSpreads

Chuan-Hsiang Han1

Department of Quantitative Finance,

NationalTsingHuaUniversity

Lei Shih

Department of Quantitative Finance,

NationalTsingHuaUniversity

1 Correspondingauthor:101,Section2,KuangFuRd.,Taiwan,30013,ROC.+886-3-5742224

chhan@mx.nthu.edu.twThisworkissupportedbyMOST104-2115-M-007-009.

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Abstract

This paper develops a modified closed-form formula for option prices under the

multifactor stochastic volatility model by means of the Fourier transform method. We

apply this result to evaluate credit spreads in the context of the structural-form

modeling. Through numerical simulation, we observe that some model parameters are

sensitive to the deformation of credit yields. This capability to generate various

shapes of the credit spread term structure enables a further study of model calibration

to corporate bond yields. With different investment grades, empirical results reveal

that the two-factor Heston model is indeed superior to other models including the

Black-Scholes model and the one-factor Heston model.

Keywords: model calibration, stochastic volatility, Fourier transform method, term structure, credit spread

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Section 1: Introduction

The structural-form approach of credit risk modeling begins with the study of

Merton (1974) that applied the option pricing theory developed by Black and Scholes

(1973). Under the constant parameter assumption, Merton (1974) derived a formula

for implied spreads which behaved too low in comparison with actual market credit

spreads. Hull and White (1987) introduced one-factor stochastic volatility models to

evaluate debt values in Merton’s (1994) framework. However, it is documented that

one-factor stochastic volatility model is often not sufficient to capture the complex

structure of time variation and cross-sectional variation.

Multi-factor stochastic volatility models are suitable to provide flexibility to

express return data such as some stylized effects, or fit the implied volatility surface.

See Christoffersen et al. (2009), Fouque et al. (2011), Han et al. (2014), Han and Kuo

(2017), and references therein. Also, in contrast to the enormous family of GARCH

family of Engle (2009) in discrete time, stochastic volatility models defined in

continuous time are natural to be applied for derivatives pricing and hedging under

the paradigms of Black, Scholes and Merton’s theory for modern finance.

Two-factor stochastic volatility models become accessible both theoretically and

empirically in recent years. Christoffersen et al. (2009) extended Heston’s (1993)

one-factor model to a two-factor stochastic volatility model built upon square root

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processes. Effects of two driving volatilities can be understood in the following. One

stochastic volatility determines the interaction between asset returns and the

“fast-varying” variance process, whereas the other stochastic volatility determines the

interaction between asset returns and the “slow-varying” variance process. Fast and

slow varying processes may correspond to impacts of instant news and economic

cycles, respectively.

Although two-factor stochastic volatility models are pronounced, it is surprising

that these models have not yet been fully explored in the credit risk literature. To fill

this gap, this article introduces a two-factor stochastic volatility specification within

the structural Merton’s (1974) model. Our contribution resides on the derivation of a

modified close-form formula for the debt value under the two-factor Heston model,

examine numerically effects of initial variances and long-run means of square root

processes, and model calibration to actual credit spreads on different investment

grades.

The rest of the paper proceeds as follows. Section 2 presents the derivation

procedure of the closed-form formula for the two-factor Heston option pricing model.

Section 3 demonstrate flexibility of the credit spread term structure. Section 4

explores several model calibrations to actual market credit spreads on different

investment grades. We conclude in Section 5.

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Section 2: Credit Spread Evaluation under Multi-factor Stochastic Volatility Models

Our objective here is to derive a modified characteristic function for the

two-factor Heston model, and its use for the credit spread evaluation under the

structural-form approach in the context of credit risk modeling.

2.1 A stylized model with stochastic volatility

Let 𝑆" ∈ ℝ, 𝑡 > 0 be the asset value process of a firm. It is defined in the

probability space Ω, ℱ" "+,, 𝒬 under a risk-neutral probability (martingale)

measure𝒬. Heston’s (1993) one factor model is one of the most popular stochastic

volatility models in the option pricing literature. It is given by

d𝑆" = 𝑟𝑆"𝑑𝑡 + 𝑉"𝑆"𝑑𝑍" (1)

d𝑉" = 𝑎 − 𝑏𝑉" 𝑑𝑡 + 𝜎 𝑉"𝑑𝑊", (2)

where 𝑉" denotes the variance process, 𝑍" and 𝑊" are Wiener processes with the

correlation ρ. The continuously compounded risk-free rate𝑟is assumed constant.

Christoffersen, Heston and Jacobs (2009) extended Heston’s (1993) model to the

following two-factor stochastic volatility model:

d𝑆" = 𝑟𝑆"𝑑𝑡 + 𝑉<"𝑆"𝑑𝑍<" + 𝑉="𝑆"𝑑𝑍=" (3)

d𝑉<" = 𝑎< − 𝑏<𝑉<" 𝑑𝑡 + 𝜎< 𝑉<"𝑑𝑊<" (4)

d𝑉=" = 𝑎= − 𝑏=𝑉=" 𝑑𝑡 + 𝜎= 𝑉="𝑑𝑊=", (5)

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where 𝑍>"𝑎𝑛𝑑𝑊>"(𝑓𝑜𝑟𝑖 = 1,2) are Wiener processes with cross variation

𝑑𝑍>"𝑑𝑊G" =𝜌>𝑑𝑡for𝑖 = 𝑗0for𝑖 ≠ 𝑗,

and 𝑊<"𝑎𝑛𝑑𝑊=" are uncorrelated. The log-return process is denoted by 𝑌" = 𝑙𝑛𝑆"

and its dynamics is governed by

d𝑌" = 𝑟 − <=𝑉<" + 𝑉=" 𝑑𝑡 + 𝑉<"𝑑𝑍<" + 𝑉="𝑑𝑍=", (6)

obtained by Ito’s formula. For analytical convenience, equations (4) and (5) used to

describe driving variances 𝑉<" and 𝑉=" are rewritten by

d𝑉<" = α< 𝑉< − 𝑉<" 𝑑𝑡 + 𝜎< 𝑉<"𝑑𝑊<" (7)

d𝑉=" = α= 𝑉= − 𝑉=" 𝑑𝑡 + 𝜎= 𝑉="𝑑𝑊=" (8)

where 𝑉> 𝑓𝑜𝑟𝑖 = 1,2 represents the long-run mean, α> denotes the rate of mean

reversion and 𝜎> the volatility of the variance process. One might further consider

three-factor stochastic volatility models. According to Molina et al. (2010) and Han

and Kou (2017), their results disclosed that the third factor’s contribution is fairly

limited and supported the use of two-factor models.

2.2 Calculate call option price by Fourier transform method

In the classical framework, the option price can be represented as an expectation

under a risk-neutral probability distribution. Consequently, its value at time t = 0 is

𝐶 𝑆,, 𝑇, 𝐾 = 𝑒UVW 𝑆W − 𝐾 XY, 𝑓 𝑆W 𝑑𝑆W, (9)

where 𝑓 𝑆W is the risk-neutral density of the underlying asset𝑆W.

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Conventionally it is easier to work with the log price rather than the price density

itself. Re-expressing equation (9) by 𝑌" = 𝑙𝑛𝑆", it is straightforward to obtain the

European call value function as

𝐶 𝑆,, 𝑇, 𝐾 = 𝑒UVW 𝑒Z[ − 𝐾 XY, 𝑓 𝑌W 𝑑𝑌W. (10)

Here we abuse the use of notation 𝑓 𝑋 by denoting the density function of the

random variable 𝑋.

The characteristic function ΨZ[ . of a given stochastic process𝑌" at time 𝑡 =

𝑇 is the Fourier transform of its probability density function 𝑓 𝑌W

ΨZ[ 𝜔 = 𝐸 𝑒>aZ[ = 𝑒>aZ[𝑓 𝑌W 𝑑𝑌W.YUY (11)

Therefore, by applying the Fourier Inversion formula, the density function of the

process𝑌W is recovered in terms of its characteristic function ΨZ[ 𝜔

f 𝑌W = <=b

𝑒U>aZ[ΨZ[ 𝜔YUY 𝑑𝜔. (12)

Moreover, based on Kendall, et al. (2009), the cumulative density function of 𝑌W is

F 𝑥 = <=− <

b𝑅𝑒

fghijkl[ a

>aY, 𝑑𝜔. (13)

The option price (10) can be further expressed by characteristic functions.

According to the main result of Heston (1993), the value of a European call option

(10) can also be expressed by

𝐶 𝑆,, 𝑇, 𝐾 = 𝑆,Φ< − 𝑒UVW𝐾Φ= (14)

where Φ<and Φ=are two probability-related quantities. We can derive Φ<and Φ=

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in terms of (13) by rephrasing

𝐶 𝑆,, 𝑇, 𝐾 = 𝑒UVW 𝑒Z[ − 𝐾Ynop 𝑓 𝑌W 𝑑𝑌W

= 𝑒UVW 𝑒Z[Ynop 𝑓 𝑌W 𝑑𝑌W − 𝐾 𝑓 𝑌W

Ynop 𝑑𝑌W

= 𝑒UVWΘ< − 𝑒UVW𝐾Θ= (15)

Comparing (14) with (15), it can be seen thatΦ= should be equal toΘ=, while Φ<

should be equal to𝑒UVWΘ< 𝑆,. We first deriveΦ=which is simply the probability of

the event defined by the log-stock price at maturity over the log-strike, i.e., ln𝑆W >

lnK:

Φ= = Θ= = 𝑃 ln𝑆W > lnK

= 1 − 𝑃 ln𝑆W ≤ lnK = 1 − F lnK

= <=+ <

b𝑅𝑒

fghixyzkxy{[ a

>aY, 𝑑𝜔 (16)

This is the derivation forΦ= in (14)

To deriveΦ<, we first calculate the integralΘ<:

Θ< = 𝑒Z[Ynop 𝑓 𝑌W 𝑑𝑌W =

fl[|}~� � Z[ �Z[

fl[|g| � Z[ �Z[

𝑒Z[YUY 𝑓 𝑌W 𝑑𝑌W

= 𝐸 𝑆Wfl[� Z[fl[|

g| � Z[ �Z[

Ynop 𝑑𝑌W

= 𝑒VW𝑆, 𝑓∗ 𝑥Ynop 𝑑𝑌W. (17)

The last equation makes use of the martingale property of 𝑒UV"𝑆". The Fourier

transforms of 𝑓∗ 𝑥 is given by

𝛹∗ 𝜔 = 𝑒>aZ[𝑓∗ 𝑌W 𝑑𝑌WYUY = � aU>

� U> (18)

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Thus, using equation (13)

Θ< = 𝑒VW𝑆,12 +

1𝜋 𝑅𝑒

𝑒U>a���Ψ���[ 𝜔 − 𝑖𝑖𝜔Ψ���[ −𝑖

Y

,𝑑𝜔

SinceΦ< = 𝑒UVWΘ< 𝑆,, we can derive

Φ< =<=+ <

b𝑅𝑒

fghixyzkxy{[ aU>

>akxy{[ U>Y, 𝑑𝜔 . (19)

This is the derivation forΦ< in (14)

In brief, we have derived the pricing formula of a European call option by

obtaining (16) and (19), then substitute these results to (14).

2.3 Calculate characteristic function of the log-price𝜳𝐥𝐧𝑺𝑻 𝝎

Crisostomo (2014) applied the characteristic function method based on the

process ln𝑆" for the one-factor Heston model. We extend this result to the

two-factor Heston model and obtain a slightly different result from Christoffersen

et al. (2009) in which the characteristic function ln ��p was used. Moreover, we

extend Crisostomo’s (2014) computational scheme to the two-factor case as well

and find that numerical results are more stable.

The characteristic function Ψ���� 𝜔 is shown follows. It can be easily

checked by a change of variable from the result of Ψno{��𝜔 derived in

Christoffersen et al. (2009).

Ψ���[ 𝜔 = 𝑒�� �,a ��X�� �,a ��X�� �,a ��X�� �,a ��X>ano �� (20)

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𝐶G 𝜏, 𝜔 =αG𝜎G=

αG − 𝜌G𝜎G𝑖𝜔 − 𝑑G 𝜏 − 2𝑙𝑛1 − 𝑒U���

1 − gG

𝐷G 𝜏, 𝜔 =αG − 𝜌G𝜎G𝑖𝜔 − 𝑑G

𝜎G=1 − 𝑒U���

1 − gG𝑒U���

gG =αG − 𝜌G𝜎G𝑖𝜔 − 𝑑GαG − 𝜌G𝜎G𝑖𝜔 + 𝑑G

dG = αG − 𝜌G𝜎G𝑖𝜔= + 𝜎G=(𝑖𝜔 + 𝜔=)

Then we can directly apply this result to the pricing framework presented in section

2.2 and calculateΦ<andΦ= as complements of two cumulative distribution functions.

2.4 Pricing credit spreads

We assume that one firm issues a zero coupon bond with a promised payment 𝐵

at maturity t = T. In this case, default occurs only at maturity with debt face value𝐵

as the default boundary. Within the Merton’s (1974) framework, the debt value

corresponding the firm at time t=0 can be expressed as:

𝐷, = 𝑆, − 𝑒UVWΕ𝒬 𝑆W − 𝐵 X (21)

That is, the debt value is equal to the difference between a firm’s asset value at time

t=0 and the European call option on the firm’s asset values with the strike price being

equal to the debt payment at maturity. To evaluate𝐷,, one should evaluate a typical

European call option as follow:

𝐶 𝑆,, 𝑇, 𝐵 = 𝑒UVWΕ𝒬 𝑆W − 𝐵 X . (22)

Taking into account the previous expressions, it is possible to express the time t=0

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debt value as:

𝐷, = 𝑆, − 𝐶 𝑆,, 𝑇, 𝐵

= 𝑆,

− 𝑆,12 +

1𝜋 𝑅𝑒

𝑒U>a���Ψ���[ 𝜔 − 𝑖𝑖𝜔Ψ���[ −𝑖

Y

,𝑑𝜔

− 𝑒UVW𝐵12 +

1𝜋 𝑅𝑒

𝑒U>a���Ψ���[ 𝜔𝑖𝜔

Y

,𝑑𝜔 ,

where the last equation is derived from equations (14, 16, 19). When a characteristic

function is chosen as (20), the two-factor Heston model is incorporated so that the

credit spread can be given by:

𝐶𝑆, = − <W𝑙𝑛 ��

�− 𝑟.

Section 3: Numerical Illustration

This section offers the numerical analysis and demonstrates how the two-factor

Heston model can generate various shapes of the term structure. Model parameter

specifications are mainly chosen from Romo (2014). We also provide a sensitivity

analysis with respect to model parameters relate to the initial variances and long-run

means.

Specifications I and II in Table 3.1 are taken from Romo (2014). Specification

III is extrapolated from the first two specifications to represent a more volatile return

process. The rest of parameters regard firm’s accounting information such as the

initial firm’s asset value 𝑆, and the debt face value B. These parameters are used to

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determine firm’s investment grades. Their values and the risk-free interest rate r are

chosen from Zhang et al. (2009).

Note that Romo (2014) only provided numerical experiments for multi-factor

stochastic volatility model specifications. Zhang et al. (2009) investigated the jump

effects with one-factor stochastic volatility models. In order to make our numerical

experiment more realistic, we choose the two-factor Heston model specifications from

Romo (2014) and each firm’s accounting information from Zheng et al. (2009).

Table 3.1: parameters specification for the two-factor Heston model

Parameter Specification I Specification II Specification III α< 1.2017 1.5141 1.9077 α= 0.3605 0.4542 0.5723 𝑉< 0.0524 0.0660 0.0831 𝑉= 0.0157 0.0198 0.0250 𝜎< 0.8968 1.1300 1.4238 𝜎= 0.2690 0.3390 0.4272 𝜌< -0.5590 -0.7043 -0.8874 𝜌= -0.1677 -0.2113 -0.2662 𝑉< 0.0581 0.0732 0.0922 𝑉= 0.0174 0.0220 0.0278

The choices of r = 0.05 and 𝑆,= 1 are fixed. With regard to different investment

grades, the debt face value is chosen as B = 0.43 corresponding to rating category A

in the study of Zhang et al. (2009); for the parametric specification II, the debt face

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value is chosen as B = 0.48 corresponding to rating category BBB of Zhang et al.

(2009); whereas for the parametric specification III, the debt face value is chosen as B

= 0.58 corresponding to rating category BB of Zhang et al. (2009).

Figure 3.1: credit spreads term structure generated under the parametric specification of Table

3.1.

The credit spreads term structure generated by the two-factor Heston model for

specification I, II, III have the similar shape to the average credit spread term structure

displayed by Zhang et al. (2009) in their sample of firms associated with the A, BBB ,

BB rating categories.

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Figure 3.2: effect of variance factor 𝑉< and 𝑉= generated by specification I of Table

3.1

Figure 3.2 displays the sensitivities associated with the variance factor 𝑉< = 𝑉<,

and 𝑉= = 𝑉=,, representing the initial variances. That is, how the change of these

initial variances reshapes the credit spread curves. One can observe that the effect of

𝑉< on the change of short-term credit spreads is more significant than the change of

long-term credit spreads. On the other hand, the effect of 𝑉= on the change of

long-term credit spreads is more significant than the change of short-term credit

spreads. These numerical examples reveal how the introduction of two volatility

factors can generate a wide range of combinations associated with short-term and

long-term patterns corresponding to credit spreads. In this sense, multifactor

stochastic volatility specifications are eligible to provide more flexibility than

single-factor models so that they can capture a wide range of shapes associated with

the term structure of credit spreads. Figure 3 displays that both𝑉< and𝑉= increase the

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credit spread, especially in the long-term, but the effect of𝑉< is more sensitive.

Figure 3.3: effect of long-run means 𝑉< and 𝑉=generated by specification I of Table 3.1

4. Calibration to market credit spreads

We have seen in the last section that the two-factor Heston model can produce

various term structures for credit spreads. In addition, the existence of a closed-form

formula is particularly useful for model calibration to actual observed market credit

spreads. The problem of model calibration aims to obtain the optimal model

parameters that are able to reproduce market credit spreads. Solving such an

optimization problem with the high dimensional parametric space is challenging.

Thus, an accurate and efficient pricing formula such as the Fourier transform are

crucial in order to obtain reliable results within a reasonable computing timeframe.

4.1 Calibration Procedure

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Throughout this paper, we choose to use the bond yield minus the risk-free

interest rate as a direct measure of credit spreads,

𝐶𝑆> = 𝑦𝑖𝑒𝑙𝑑> − 𝑟.

We calibrate a series of models — Black and Scholes, one factor (Heston) stochastic

volatility, two factor (Heston) stochastic volatility — across rating categories of high

investment grade (A+), low investment grade (AA-), and speculative grade (BBB).

The goal of calibration is to search for the optimal parameter set that minimizes

the distance measure between model predictions and observed market prices. Under a

risk-neutral measure, the two-factor Heston model is equipped of ten unknown

parameters Ω = α<, α=, 𝑉<, 𝑉=, 𝜎<, 𝜎=, 𝜌<, 𝜌=, 𝑉<, 𝑉= . The procedure of model

calibration has two folds. Firstly, define a measure to quantify the distance between

theoretical model values and observed market prices. Secondly, execute an

optimization scheme to determine the parameter values that minimize such a distance

measure. Our distance measure is defined by the mean sum of squared error ratios

Ι Ω =1𝑁

𝐶𝑆£> 𝐵, 𝑇 − 𝐶𝑆¤¥¦> 𝐵, 𝑇𝐶𝑆¤¥¦> 𝐵, 𝑇

=

,§

>¨<

where 𝐶𝑆£> 𝐵, 𝑇 denotes the i-th theoretical model credit spread using the

parameter setΩ, 𝐶𝑆¤¥¦> 𝐵, 𝑇 denote the i-th observed market credit spreads.

4.2 Calibration Data and Results

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We take yield spreads from market prices of corporate bonds for three firms

including IBM on 27 May 2016, when it was rated A+, and HP, a firm rated BBB, on

12 June 2016, and Chevron, an energy firm rated AA-, on 14 May 2016. The spreads

are obtained from Morningstar (www.morningstar.com.)

A yield is a value to describe a zero coupon bond in the bond market. Since the

market data consists of corporate coupon bonds, we need to employ the bootstrapping

method to construct a zero-coupon fixed-income yield curve from those

coupon-bonds. This method is based on the assumption that the theoretical price of a

bond is equal to the sum of the cash flows discounted at the zero-coupon rate of each

flow. Its implementation is given below.

Zero-coupon bond yields can be retrieved from the following formula (for the

bond that pays dividends):

𝑃 = 𝑁

(1 + 𝑦𝑡𝑚o)W+

𝐶>(1 + 𝑦𝑡𝑚>)"h

oU<

>¨<

,

where 𝐶> is the zero coupon bond price,𝑦𝑡𝑚> is the zero coupon rate of 𝑡>-year bond,

N is face value. The risk-free rate data used in our calibration are the U.S treasury

yields for three months, six months, one year, to ten years.

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Figure 4.1: Comparisons of fitting IBM yield spreads by Black-Scholes, one-factor

and two-factor stochastic volatility models. The short rate is fixed at r = 0.0025.

Debt-to-asset ratio is B/𝑆,=0.36.

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Figure 4.2: Comparisons of fitting HP yield spreads by Black-Scholes, one-factor and

two-factor stochastic volatility models. The short rate is fixed at r = 0.0025.

Debt-to-asset ratio is B/𝑆,=0.27.

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Figure 4.3: Comparisons of fitting Chevron yield spreads by Black-Scholes,

one-factor and two-factor stochastic volatility models. The short rate is fixed at r =

0.0025. Debt-to-asset ratio is B/𝑆,=0.16.

Figure 4.1, 4.2, and 4.3 show the curve fitting of the actual credit yield spreads by three different models. The two-factor Heston model is visualized to produce various term structures that are closest to actual market data. Tables 4.1-4.3 records estimated model parameters. Table 4.4 describes the fitting error (MSE) of three models. For each firm, MSE of the two-factor stochastic volatility model is smallest.

However, we should point out related problems such as over fitting and

parameter identification. The usual way to treat the over-fitting problem is to perform sensitivity analysis and/or robustness check for out-of-sample test. We have done some sensitivity analysis in section 3 and seen that credit spread curves can be sensitive to the change of some volatility model parameters such as long-run mean and initial variances. These are done through numerical experiments. As for empirical

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test, we encounter a problem about data. Since the corporate bond prices in the whole term structure are not daily observable in our database, it becomes difficult to perform an out-of-sample test. Zhang et al. (2009) only compared several models via calibration to CDS prices and didn’t perform the out-of-sample test as well. As for the identification problem, Han and Kuo (2017) discovered such problem under a two-factor exponential OU (Ornstein-Uhlenbeck) stochastic volatility model. It happened that not every model parameter can be uniquely identifiable in such high-dimensional and complex model structure. Nevertheless, most literature recognize that multi-factor stochastic volatility models are essential to account for economic or business cycles of short-term and long-term change of the market risk.

Table 4.1: parameters specification of the Black-Scholes model for each firm’s credit

rating

Parameter IBM(A+) HP(BBB) Chevron(AA-) 𝑉 0.301295 0.500082 0.444667

Table 4.2: parameters specification of one-factor stochastic volatility model for each firm’s credit rating

Parameter IBM HP Chevron 𝑉 2.742524 2.433317 5 𝑉 0.074364 0.202235 0.129696 𝜎 1.778405 1.700618 2.033256 𝜌 0.36894 0.525417 -1 α 21.26858 7.150363 15.93781

Table 4.3: parameters specification of two-factor stochastic volatility model for each firm’s credit rating

Parameter IBM HP Chevron 𝑉< 0.122286 0.077631 3.93657 𝑉= 1.557314 1.559366 2.904554 𝑉< 0.103331 0.340766 0.019903 𝑉= 0.002996 0.009473 0.108046 𝜎< 0.366838 0.637395 0.578341 𝜎= 0.281581 0.331587 1.824795 𝜌< 0.998741 0.966932 -0.85514 𝜌= -0.9905 -0.93536 -0.999

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α< 0.651262 0.616602 17.99047 α= 13.32973 5.891756 17.99455

Table 4.4: mean square error specification of three models for each firm’s credit

rating

MSE IBM(A+) HP(BBB) Chevron(AA-) Black-scholes 0.0224 0.5886 0.1833

One-factor 0.0161 0.4261 0.0039 Two-factor 0.0029 0.2037 0.0019

Section 5: Conclusion

This paper provides a modified closed-form formula by means of the Fourier

transform method for the multifactor stochastic volatility option pricing model.

Through numerical experiments, the two-factor Heston model demonstrates flexibility

to model the credit spread term structure. It is peculiarly observed that some model

parameters are sensitive to the deformation of short-term credit spread, while others

are sensitive to long-term credit spreads.

Two aforementioned advantages, including (1) accurate and fast calculation by

the closed-form solution, and (2) flexible shapes of the credit spread term structure,

enables the study of model calibration Within the diffusion family, three

models—Black and Scholes (BS), one factor stochastic volatility (1SV), two factor

stochastic volatility (2SV)— are introduced to distinct rating categories of high

23

investment grade (A+), low investment grade (AA-), and speculative grade (BBB).

Empirical studies confirm that two-factor Heston model best fit to the credit yields

across these corporate bonds.

24

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